Post on 02-May-2018
Asian Economic and Financial Review, 2016, 6(5): 277-297
277
† Corresponding author
DOI: 10.18488/journal.aefr/2016.6.5/102.5.277.297
ISSN(e): 2222-6737/ISSN(p): 2305-2147
© 2016 AESS Publications. All Rights Reserved.
A COMPARATIVE STUDY OF THE TAIWAN AND JAPAN EQUITY AND FOREIGN EXCHANGE MARKETS: MODELING, ESTIMATION AND APPLICATION OF THE COMPONENT GARCH-IN-MEAN MODEL
Hsiang-Hsi Liu1† --- Robin K Chou2
1 Distinguished Professor, Graduate Institute of International Business, National Taipei University, Taipei, Taiwan 2
Professor, Department of Finance, National Chengchi University, Taipei, Taiwan
ABSTRACT
The main purpose of this paper is to verify the effectiveness of the bivariate Component GARCH-in-mean (GARCH-
M) model and analyze the interactions and risk premium of equity markets by exploring the short- and long-run
volatility components on both the Taiwanese and Japanese equity markets. We show that unexpected shocks of
volatility will in general influence the fluctuations of both equity and foreign exchange markets. Persistence on the
long-run volatility components of both markets is also found. The results also reveal that the positive risk-return
relation on equity markets can be further verified when the impacts of short and long-run volatility components are
decomposed by the Component GARCH-M model. The decomposition can also facilitate reflecting the transitory and
permanent volatility impacts of foreign exchange exposure on the returns of equity markets.
© 2016 AESS Publications. All Rights Reserved.
Keywords: ARCH, Component GARCH-in-mean model (GARCH-M), Risk premium, Foreign currency exposure, Equity market, Transitory
and permanent volatilities.
JEL Classification: G1; C32; F31.
Contribution/ Originality
This study is one of very few studies which have investigated risk premium and the interactions between equity
markets, by exploring the short- and long-run volatility components on both the Taiwanese and Japanese equity and
foreign exchange markets via estimating bivariate Component GARCH-in-mean (GARCH-M) mode.
1. INTRODUCTION
It is widely believed that exchange rates affect the value of assets. Exchange rates are one of the major sources of
uncertainty, being that they are typically four times as volatile as interest rates. The analysis of the correlation
between equity returns and foreign exchange rates has received much attention in financial econometric studies in
recent years. The ways of modeling these financial time series have been enriched by standard GARCH processes.
However, when capturing the long-term and short-term volatility effects, how to choose an appropriate model is a
complicated problem. It has been observed in the literature that volatility effects may be better modeled using a
Component GARCH-M model. To consider the permanent and transitory effects of the impact on rate of returns and
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the effects of short- and long-run volatility components on both Taiwanese and Japanese equity markets, this paper
adopts the Component GARCH-M model developed by Engle and Lee (1999).
Considering that Taiwan and Japan have a close interaction both economically and financially In South-East
Asia, Taiwan and Japan’s economic model may play a part in their respective exchange rate risk. It is important to
note that Taiwan and Japan have a close relationship on trades and investments, and thus exchange rates can
potentially influence both side’s capital and equity markets. And furthermore reduce the risk reduction for portfolio
investment. The purpose of this paper is to verify the effectiveness of the bivariate Component GARCH-M model and
analyze the risk premium of equity markets by exploring the effects of the short and long-run volatility components
on both Taiwanese and Japanese equity markets. Under the Component GARCH-M framework, time series effects of
volatility are decomposed into short and long-run volatility components. The estimated model reveals that there is a
rich interaction between the rates of return for both the Taiwan and Japan equity markets. The Component GARCH-
M model appropriately captures the effect of foreign currency exposure on the volatility of equity markets.
Furthermore, it also better delineates the effect of equity shocks on the volatility of equity markets.
This paper is organized as follows. The literature review is presented in section 2, followed by section 3, where a
theoretical framework of the Component GARCH-M model is presented. The data and variables definitions are
described in section 4. Section 4 also presents empirical results of this study. Section 5 concludes this article.
2. LITERATURE REVIEW
Traditional econometric models have a constant one-period forecast variance. Engle (1982) generalized this setup
and develops a new class of econometric processes called autoregressive conditional heteroscedastic (ARCH)
processes. Time-varying volatility is not only important in forecasting future market movements but is also central to
a variety of financial issues, such as portfolio design, risk management, and derivative pricing. Although it is
common to find a significant statistical relationship between current volatility and lagged volatilities, it is generally
difficult to decide the exact models that appropriately capture the inter-temporal dependencies observed on the data
series. Bollerslev (1986) generalizes the ARCH framework introduced by Engle (1982) into the GARCH framework,
where it incorporates past conditional variances into current conditional variance, In his study, stationary conditions
and autocorrelation structures of the GARCH model for this new class of parametric models are derived, furthermore,
maximum likelihood estimation and testing procedures are also considered.
Engle et al. (1987) continue to extend the ARCH model into the ARCH-in-Mean (ARCH-M) model, in which the
conditional mean is an explicit function of the conditional variance of the ARCH-M process. Since the ARCH in
mean model can deal with the trade-off of the risk-return profile in finance research, it causes a boom of finance
research on the time-varying conditional variance. When the volatility of financial asset returns changes, agents
naturally suspect that time-varying risk premia may be operative. Later on Lee (1999) also applies the multivariate
component model for conditional asset return covariance, which is an extension of the invariant volatility component
model. The conditional covariance is divided into a long-run (trend) component and a short-run (transitory)
component. Through the decomposition, the long-run correlation and volatility co-persistence can be studied
separately. Through their research, Christoffersen et al. (2008) proposed a model with the volatility of returns
consisting of two components. One is a long-run component, which can be modeled as fully persistent, and the other
is a short-run component that has a zero mean. The model can be viewed as a refined version of Engle and Lee (1999)
allowing for easy valuation of European options.
Watanabe and Harada (2006) examine the effects of Bank of Japan's (BOJ) intervention on the volatility and the
level of the yen/dollar exchange rate. Specifically, the conventional GARCH model and the component GARCH
model, where the volatility consists of short-run and long-run components, are estimated using the BOJ's and the
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Federal Reserve’s system intervention data. Results based on the component GARCH model show new evidence on
the effects of the BOJ's intervention on the volatility of the yen/dollar exchange rate. Yau and Nieh (2009) investigate
the exchange rate effects of the New Taiwan Dollar against the Japanese Yen on stock prices in Japan and Taiwan by
the threshold error correction model (TECM). The finding suggests that there is a long run equilibrium relationship
between NTD/JPY and the stock prices of Japan and Taiwan. However in the short run, the results of TECM Granger
causality tests show that no short run causal relationships exist between the two financial assets considered for both
countries` cases. Recently, Lee et al. (2010) adopt the Component GARCH model to examine the relationships
between volatility and trading volume in five major Asian index futures markets. Their empirical results confirm that
the volatility of stock index futures in Asian markets includes permanent and transitory components. Also, the
information conveyance in these markets is not very efficient.
In general, regarding emerging markets around the East-Asia region not quite much research has been done on
the interactive effects of the stock and foreign exchange markets regarding the view of the short- and long-run
volatility components by the Component GARCH model. Some researches refer to the connections between the stock
and foreign exchange markets through the spillovers of market volatility by GARCH-type models or via linear or
even nonlinear co-integration analysis. Even though both the GARCH model (focused on market volatility spillovers)
and the co-integration analysis (focused on the co-integration vector variation of the long-term correlation after
exogenous chocks), these methods still cannot help to detect the impacts of information transmission on the
permanent and transitory return volatility between the stock and foreign exchange markets. In light of such short- and
long-run characteristics of return volatility, our study applies the Component GARCH model, modified from the
traditional GARCH model by Engle and Lee (1999) that decomposes conditional variance into permanent and
transitory components.
3. THEORETICAL FRAMEWORK
3.1. The Framework of Component GARCH Model
To explore the short- and long-run volatility components and interactions on Taiwanese and Japanese equity with
foreign exchange markets, this article proposes an application and extension of the univariate Component GARCH
model developed by Engle and Lee (1999). The framework of the Component GARCH model is as follows:
ttt xy ),0(~| 1 ttt hN (1)
1
2
1 ttt hh (2)
Where ty is a covariance-stationary process; tx includes the regressors; t is the information set available at
time t; and th is a conditional variance measurable in respect to 1t . The multi-step-ahead forecast of conditional
variance is denoted as )|( 1 tktkt yVarh . Assuming 1 , the multi-step-ahead forecast of conditional
variance under GARCH (1, 1) framework is
11k
kth
1 as k (3)
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Which will converge to unconditional variance )(2
thVar (Bollerslev, 1986). Therefore, the GARCH (1, 1)
framework can be rewritten as
1
2
1
21 ttt hh
2
1
22
1
2 tt h (4)
Where the expected values of the last two terms are zero.
The Component GARCH model permits both a long-run volatility component of tq and a short-run volatility
component of )( tt qh . The long-run volatility component is slowly mean reverting, whereas the short-run
component is more volatile. Following Engle and Lee (1999) the conditional variance of Component GARCH model
is defined as follows:
111
2
1 tttttt qhqqh (5)
1
2
11 tttt hqq (6)
It is clear that 2 in expression (4) is replaced with tq in expression (5) and 1
2
1 tt h is the prediction
error of tq in expression (6).
Define ktq to be the multi-step-ahead forecast of long-run component tq under information set 1t :
tkt qkq 0k (7)
Then the multi-step-ahead forecast of short-run component )( tt qh is
tt
k
ktktktkt qhqhqh 11 (8)
If 1 and )( tt qh is a stationary )1(AR process, then the value of multi-step-ahead forecast of short-run
component )( tt qh tends to zero when k approaches infinity:
0 ktkt qh as k (9)
Therefore, the conditional variance of the Component GARCH model will be equal to its long-run component in
the long-run.
3.2. The Property of Component GARCH Model
To incorporate the non-unit root process into the framework, the Component GARCH model can be generalized
as follows:
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111
2
1 tttttt qhqqh (10)
1
2
11 tttt hqq (11)
As long as , tq indicates that the conditional variance would exhibit the behavior of long-term
memory. The multi-step-ahead forecast of short- and long-run component under a more generalized Component
GARCH framework can be obtained as follows:
tt
k
ktkt qhqh (12)
t
kk
kt qq ]11[ (13)
The long-run volatility component will diminish faster than the short-run component when 1 , 1)(
, and )( . The conditional variance will converge to a constant when the long-run volatility component is
stationary:
1ktkt qh as k (14)
To examine the relationship between Component GARCH and GARCH models, substituting expression (11)
into expression (10) obtains
2
2
2
11 ttth
1 2t th h (15)
Thus, GARCH (2, 2) describes the conditional variance of Component GARCH framework. Distinguishing
short- and long-run volatility components enables Component GARCH framework to appropriately depict volatility
dynamics. The Component GARCH framework will be degenerated into GARCH (1, 1) model when α=β=0 or ρ=
=0 in expression (15). Therefore, the GARCH (1, 1) model only portraits plain dynamic relationships of
conditional variance through decomposition and component GARCH framework will portrait volatility dynamics
better than GARCH (1, 1) model.
Defining the prediction error tv tt h2
, expression (15) can be rewritten as
2
2
2
1
2 1 ttt
21 ttt vvv (16)
The squared innovation of Component GARCH framework follows an ARMA (2,2) process. We obtain the
following through expression (16):
2
2
2
1
2
1
2 1 tttt
21 ttt vvv (17)
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Here it follows an ARIMA (1, 1, 2) process when ρ= l. The GARCH (1,1) model with α=β=0 could be
characterized by an ARIMA (0,1,1) process. Analogous to expressions (5) and (6), the previous expressions can be
obtained:
11
2
1
2
1 tttttt vvqq (18)
11 ttt vqq (19)
The maximum likelihood method is used to estimate the parameters of the variance process due to the
heterogeneity of error terms. The GARCH process is a covariance-stationary process if and only if the sum of its
dynamic parameters is less than one (Bollerslev, 1986). Under Component GARCH framework, the conditions of
stationary implies
11 (20)
The stationary conditions will achieve when ρ<1 and <1. If both short- and long-run volatility
components are covariance-stationary, the conditional variance also is covariance-stationary.
3.3. Volatility Persistency Test and Explanation
If the long-run component of conditional variance exhibits a unit-root process, i.e., an IGARCH (2, 2) the impact
of innovations on volatility will exhibit a persistent effect (Engle and Bollerslev, 1986). Therefore, we need to
explore the impact effect of volatility under Component GARCH framework. The following comparative statics can
be obtained from expressions (10) and (11):
2
1ttq (21)
2
1ttt qh (22)
Therefore, and α portrait the effects of unexpected shocks from short and long-run components of conditional
variance, respectively. The multi-period-ahead effects of a past innovation on short and long-run volatility
components are
k
tt
k
tkt qq
2
1
2
1 (23)
k
ttt
k
tktkt qhqh
2
1
2
1 (24)
The impact effect on volatility components will slowly decay when 1>ρ> . If 1 , the
impact effect on short-run volatility components will gradually vanish. However, the impact effect on long-run
volatility components will persist:
2
1
2
1 tttkt qq (25)
Under the Component GARCH framework, the Lagrange-multiplier and likelihood ratio tests can be used to test
the persistence of volatility.
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3.4. Non-Negativity of Conditional Variance under Component GARCH Framework
To explore the non-negativity of conditional variance under the Component GARCH framework, substituting
expression (10) into expression (11) obtains
1
2
11 tttt hqh (26)
In which th follows time-varying GARCH (1, 1) process. If 1>ρ> >0 , β> >0, α>0, 0 , >0, and
>0, the conditional variance th maintains its non-negativity, so does the long-run volatility component tq .
Substituting expression (10) into expression (11) tq can be simplified to GARCH (2, 2) process in terms of past
squared innovations:
1
2
2
2
11 tttt qq 2 tq (27)
Under the Component GARCH framework, the non-negativity of long-run volatility component will be achieved
when 1>ρ> >0, β> >0, α>0, 0 , >0, and >0 (Nelson and Cao, 1992).
4. EMPIRICAL ANALYSIS
We explore the interactions between equity and exchange markets for Taiwan and Japan based on the
econometric models presented above. The main purpose here is to examine the goodness of fit of the Component
GARCH-M models for our data. Meanwhile, we will test the equity and foreign exchange risk impacts on the
expected returns of equity market. Firstly, we briefly describe the characteristics of the data. Secondly, we present the
results of unit root tests, serial correlation tests and ARCH effect tests on our data. Thirdly, the empirical model is
established. Finally, the empirical results of the Component GARCH-M models are presented.
4.1. Data Characteristics
We collected data for Taiwan’s weighted equity index and daily exchange rate data as well as for the Japan
Nikkei 225 equity index and daily exchange rate data. All data are daily close price and extracted from the
Datastream database. The sample period is from January 1, 1992 to December 31, 2011. Before the year 2000, in
Taiwan Saturdays were also trading days, meaning a total of six trading days per week; however, equity and
exchange markets in Japan are closed on Saturdays. To match the data, we deleted Saturday’s trading data in Taiwan
before 2000. In general, investors focus more on asset returns than on prices. From the viewpoints of econometrics,
the price series are not stationary. Hence, it is necessary to transform the price series into returns series. Traditionally,
we can achieve this by taking logarithmic and making first difference for the consecutive prices, i.e; Ri,t = (ln Pi,t –
lnPi,t-1). Where Ri,t denotes the return at the period t for the i-th country; and Pi,t is the closing price at the t day for the
i-th country. As for the foreign exchange market, the same method is applied for the closing price of exchange rates,
letting ri,t = (ln pi,t – ln pi, t-1). We distinguish the data of foreign exchange markets from equity markets with lower
case letters. Other than this, the definitions of variables are similar to the equity markets.
We summarize in Table 1 the basic statistics of the raw data for equity and foreign exchange markets in Taiwan
and Japan. Returns data for equity markets and foreign exchange markets in Taiwan and Japan also are outlined in
Table 2. In Table 1, both Taiwanese equity and foreign exchange markets display positive skewness. However, both
Japanese equity and foreign exchange markets are
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Table-1. Summary statistics for equity prices and foreign exchange rates in Taiwanese and Japanese
markets
Equity Market Foreign Exchange Market
Taiwan Japan Taiwan Japan
Mean 6322.983 15181.33 30.783 112.638
Std dev 1528.443 3906.092 3.2186 11.261 Max 10202.200 23801.000 35.165 147.210
Min 3135.560 7054.980 24.507 80.620
Skewness 0.265 - 0.179 0.497 - 0.065
Kurtosis 2.634 1.894 1.723 3.016 Jarque-Bera 141.353*** 249.195*** 500.511*** 143.403***
)24(Q 71,324.000*** 72,044.000*** 74,104.000 *** 70,112.000*** Note: *** represents significant at 1% level.
Table-2. Rate of returns of equity prices and foreign exchange rates for Taiwanese and Japanese markets
Equity Market Foreign Exchange Market
Taiwan Japan Taiwan Japan Mean 0.005 - 0.008 0.002 - 0.003 Std dev 0.673 0.679 0.114 0.311
Max 2.833 5.748 1.470 2.786 Min - 3.029 - 5.260 - 1.165 - 2.773
Skewness - 0.112 0.108 1.295 - 0.419
Kurtosis 4.955 8.057 32.284 8.993 Jarque-Bera 768.037*** 4724.693*** 165184.600*** 7206.899***
)24(Q 74.124*** 31.825*** 308.812*** 38.421*** )24(2Q 532.810*** 460.115*** 984.227*** 794.134***
794.134***
Note: *** represents significant at 1% level.
Negatively skewed. Meanwhile, the Jarque-Bera statistics show that the data are not normally distributed.
Finally, autocorrelation tests (Q(24)) also reject the data being white noise. This shows that it is important to take the
first difference for stationarity.
We display the summary statistics for returns in Table 2. For our sample period, the returns for equity and
foreign exchange markets in Taiwan are positive. However, they are both negative for equity and foreign exchange
markets in Japan. As for skewness, the Taiwanese exchange market and Japanese equity market appear to be positive
skewed and other markets are negatively skewed. For the fourth moments (kurtosis) of the series, we find equity
markets and foreign exchange markets for these two countries to be more than 3. Namely, the data for equity and
exchange markets possesses fat tails. From Table 2, the data do not appear to be normally distributed from the J-B
statistics.
4.2. Unit Root, Serial-Correlation and ARCH Effect Tests
The early and pioneering work on testing for a unit root in a time series was done by Dickey and Fuller (DF).
Subsequently, Phillips and Perron (PP) developed a more comprehensive theory on the unit root non-stationarity. The
tests are similar to the augmented Dickey-Fuller (ADF) tests, but they incorporate an automatic correction to the DF
procedure to allow for auto-correlated residuals.
If we find that the data does possess a unit root, it means that the series is not stationary. One can use first order
difference to remedy the stationarity. After taking first order difference and changing the time series into a stationary
one, it is said to be integrated to the order of one, i.e., denoted I (1). In Tables 3, 4 and 5, we present the results for the
unit root test using the ADF test and the PP test, respectively. No matter which functional type is used for the original
data, namely the constant term and/or time trend, all of their test statistics are lower than the critical value. Therefore,
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the price series and exchange rates are not stationary. However, after the first order differencing, all data follows a
stationary process based on conventional levels of significance.
We further test for series correlation. It is conventional to fit the auto-regressive model for the return’s mean
equation. However, these models have to satisfy no serial correlation for the residual terms and follow white noise.
For this purpose, we introduce Ljung-Box Q statistics developed by Ljung and Box. Meanwhile, the index of AIC
and SBC are incorporated for the selection of appropriate lag periods. All of the results are presented in Tables 6 to 8.
Basically, the optimal number of lag terms in the conditional mean equation is determined by the likelihood ratio test,
the results show that optimal lag length for the Taiwanese stock market is 4. And for the Taiwanese foreign exchange
market, Japanese stock and foreign exchange market, the optimal lag lengths are 2, based on the values of AIC/BIC
(Table 6, 7 and 8) under Ljumg–Box Q test.
Table-3. Unit root testing results of ADF and PP tests
Drift and Trend Drift W/o Drift and Trend
ADF
Taiwan Equity (5) -1.815 -1.982 -1.521
Japan Equity (3) -2.410 -1.201 -1.082
Tw Exch. Rate (3) -1.924 -0.801 -1.411
J. Exch. Rate (3) -2.201 -2.101 -0.358
PP
Taiwan Equity (5) -1.945 -1.982 -1.952
Japan Equity (3) -2.582 -1.701 -1.705
Tw Exch. Rate (3) -2.105 -0.902 -1.421
J. Exch. Rate (3) -2.055 -2.123 -2.215 Note: The number of lag periods is in parenthesis.
Table-4. The ADF and PP test results for logarithm of equity prices and exchange rates
Drift and Trend Drift W/o Drift and Trend
ADF
Taiwan Equity (5) -1.852 -2.145 -1.576
Japan Equity (3) -1.812 -0.464 -1.831
Two Exch. Rate (3) -1.802 -0.785 -1.412
J. Exch. Rate (3) -2.101 -2.012 -0.711
PP
Taiwan Equity (5) -1.953 -1.988 -1.648
Japan Equity (3) -2.301 -1.384 -1.925
Tw Exch. Rate (3) -2.162 -0.991 -1.546
J. Exch. Rate (3) -2.114 -1.041 -0.615 Note: The number of lag periods is in parenthesis
Table-5. ADF and PP tests results for the equity returns and the rate of returns of exchange rates
Drift and Trend Drift w/o Drift and Trend
ADF
Taiwan Equity (5) -25.815*** -25.810*** -25.161 ***
Japan Equity (3) -34.112 *** -34.152 *** -34.168***
Tw Exch. Rate (3) -25.710 *** -25.812 *** -25.144 ***
J. Exch. Rate (3) -32.452 *** -32.115 *** -33.699 ***
PP
Taiwan Equity (5) -55.112 *** -55.912 *** -54.612***
Japan Equity (3) -58.143 *** -58.321 *** -58.415 ***
Tw Exch. Rate (3) -69.112 *** -69.665 *** -69.713 ***
J. Exch. Rate (3) -57.623 *** -57.182*** -57.145 *** Note: *** represents significant at 1% level. The number of lag periods is in parenthesis.
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Table-6. The optimal number of lag periods for equity markets
No
Taiwan Equity Japan Equity
AIC BIC AIC BIC
1 -169.596 -155.022 -1537.785 -1387.723
2 -170.762 -149.152 -1541.433*** -1388.335***
3 -169.863 -149.127 -1539.598 -1377.464
4 -175.868*** -148.188*** -1539.598 -1377.464
5 -174.195 -141.479 -1536.243 -1362.029
6 -172.027 -134.274 -1535.987 -1355.747
Note: *** represents significant at 1% level.
Table-7. Optimal number of lag periods for foreign exchange markets
Tw Exch. Rate J. Exch. Rate
AIC BIC AIC BIC
1 -9790.187 -9968.056 -5923.128 -5850.999
2 -9800.945*** -9872.780*** -5924.317*** -5843.152***
3 -9799.024 -9864.823 -5919.631 -5835.430
4 -9803.647 -9863.410 -5918.182 -5827.956
5 -9849.565 9903.288 -5916.361 -5828.090
6 -9848.139 -9895.831 -5916.260 -5813.953 Note: *** represents significant at 1% level.
Table-8. Residual tests of conditional mean equations for Taiwanese and Japanese equity and foreign
exchange markets
Equity Market Foreign Exchange Market
Taiwan Japan Taiwan Japan
Q(8) 1.832 2.512 2.657 3.612
Q(16) 15.012 10.682 13.778 13.672
Q(24) 21.116 25.195 24.621 23.118 2Q (8) 403.665*** 173.812*** 426.334*** 82.512***
2Q (16) 603.182*** 255.812*** 631.578*** 109.612***
2Q (24) 800.565*** 324.417*** 840.415*** 198.201***
Note: *** represents significant at 1%.
Finally, we show the results for ARCH effect. We perform the ARCH-LM test to determine whether an ARCH-
effect is present in the residuals of an estimated model, such as AR (p). We gather the AR (p)’s residuals, square the
residuals, and then regress them on their own lags to test for ARCH to the order of p. We can obtain R2 from this
regression. The test statistics is defined as TR2 from regression where T is the total number of observations, and is
distributed as a chi-squared distribution with the p degree of freedom. We describe these results in Table 9. It is clear
that these return series exhibit ARCH effects. Hence, it is appropriate to fit a GARCH model for further analysis.
Table-9. ARCH-LM tests of squared residuals of conditional mean equations for Taiwanese and
Japanese equity and foreign exchange markets
Equity Market Foreign Exchange Market
Lag Taiwan Japan Taiwan Japan
1 631.522*** 652.471*** 391.885*** 293.645***
2 661.801*** 640.471*** 392.732*** 305.852***
3 668.150*** 665.114*** 516.551*** 307.152***
4 690.150*** 675.114*** 625.401*** 310.402***
5 682.11*** 674.801*** 630.712*** 310.614***
Note: *** represents significant at 1% level.
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4.3. Model Setups for Empirical Study
To understand the interactions of transitory and permanent risks on equity and foreign exchange markets, the
variable of return volatility is incorporated as a regressor into the conditional mean equation. The process can provide
the Information about risk premium in the short- and long-run situation. To build up various Component GARCH-M
models, we first define the relevant variables and parameters, and then construct the univariate Component GARCH-
M models to conduct our empirical analyses.
The relevant variables and parameters for various univariate Component GARCH-M models are defined as
follows:
RS and RE represent the Taiwanese equity and foreign exchange markets, respectively;
RS
ty and RE
ty represent the rate of returns of the Taiwanese equity and foreign exchange markets in period t,
respectively;
RS
t and RE
t represent the innovations of the Taiwanese equity and foreign exchange markets in period t,
respectively;
RS
th and RE
th represent the conditional variance of the rate of returns of the Taiwanese equity and foreign exchange
markets in period t, respectively;
RS
tq and RE
tq represent long-run volatility component of the Taiwanese equity and foreign exchange markets in
period t, respectively;
RS
ts and RE
ts represent short-run volatility component of the Taiwanese equity and foreign exchange markets in
period t, respectively, where ttt qhs ;
212121 ,,,,,,,,,,,, dddccc and 2 are relevant parameters to be estimated in various univariate
Component GARCH-M models.
The corresponding variables and parameters for the Japanese equity and foreign exchange markets are similarly
defined with RS and RE replaced by JS and JE respectively.
There are three Component GARCH-M models considered in this study. The first Component is the GARCH-M
model which is constructed to investigate the impact of endogenous equity volatility and exogenous foreign exchange
volatility on the rate of returns in equity markets and also to explore the effects of the short and long-run volatility
components on both Taiwanese and Japanese equity markets. Cases 1 and 2 characterize the corresponding
conditional mean and variance equations for Taiwanese and Japanese equity markets examined in the first
Component GARCH-M model:
CASE 1: Impact of endogenous equity volatility
4
1
0
k
RS
t
RS
t
RSRS
kt
RS
k
RSRS
t hcyaay
RS
t
RS
t
RSRS
t
RS
t
RSRS
t
RS
t qhqqh 111
2
1
Asian Economic and Financial Review, 2016, 6(5): 277-297
288
© 2016 AESS Publications. All Rights Reserved.
RS
t
RS
t
RSRS
t
RSRSRS
t hqq 1
2
11
2
1
0
k
JS
t
JS
t
JSJS
kt
JS
k
JSJS
t hcyaay
JS
t
JS
t
JSJS
t
JS
t
JSJS
t
JS
t qhqqh 111
2
1
JS
t
JS
t
JSJS
t
JSJSJS
t hqq 1
2
11 (28)
CASE 2: Impact of exogenous foreign exchange volatility
4
1
0
k
RS
t
RE
t
RERS
kt
RS
k
RSRS
t hcyaay
RS
t
RS
t
RSRS
t
RS
t
RSRS
t
RS
t qhqqh 111
2
1
RS
t
RS
t
RSRS
t
RSRSRS
t hqq 1
2
11
2
1
0
k
JS
t
JE
t
JEJS
kt
JS
k
JSJS
t hcyaay
JS
t
JS
t
JSJS
t
JS
t
JSJS
t
JS
t qhqqh 111
2
1
JS
t
JS
t
JSJS
t
JSJSJS
t hqq 1
2
11 (29)
The second Component GARCH-M model is constructed to investigate the impacts of the short- and long-run
volatility components of endogenous equity volatility and exogenous foreign exchange volatility on returns of equity
markets and to explore the effects of the short and long-run volatility components on equity markets. Cases 3 and 4
characterize the corresponding conditional mean and variance equations in the second Component GARCH-M model:
CASE 3: Impact of the short- and long-run volatility components of endogenous equity volatility
4
1
210
k
RS
t
RS
t
RSRS
t
RSRS
kt
RS
k
RSRS
t scqcyaay
RS
t
RS
t
RSRS
t
RS
t
RSRS
t
RS
t qhqqh 111
2
1
RS
t
RS
t
RSRS
t
RSRSRS
t hqq 1
2
11
2
1
210
k
JS
t
JS
t
JSJS
t
JSJS
kt
JS
k
JSJS
t scqcyaay
JS
t
JS
t
JSJS
t
JS
t
JSJS
t
JS
t qhqqh 111
2
1
JS
t
JS
t
JSJS
t
JSJSJS
t hqq 1
2
11 (30)
CASE 4: Impact of the short- and long-run volatility components of exogenous foreign exchange volatility
Asian Economic and Financial Review, 2016, 6(5): 277-297
289
© 2016 AESS Publications. All Rights Reserved.
4
1
210
k
RS
t
RE
t
RERE
t
RERS
kt
RS
k
RSRS
t scqcyaay
RS
t
RS
t
RSRS
t
RS
t
RSRS
t
RS
t qhqqh 111
2
1
RS
t
RS
t
RSRS
t
RSRSRS
t hqq 1
2
11
2
1
210
k
JS
t
JE
t
JEJE
t
JEJS
kt
JS
k
JSJS
t scqcyaay
JS
t
JS
t
JSJS
t
JS
t
JSJS
t
JS
t qhqqh 111
2
1
JS
t
JS
t
JSJS
t
JSJSJS
t hqq 1
2
11 (31)
The third Component GARCH-M model is constructed to investigate the impact of foreign exchange volatility
from the previous period and the short- and long-run volatility components of foreign exchange volatility from the
previous period on returns of equity markets and to explore the effects of the short- and long-run volatility component
on equity markets. Cases 5 and 6 characterize the corresponding conditional mean and variance equations in the third
Component GARCH-M model:
Case 5: Impact of exogenous foreign exchange volatility in the previous period
4
1
0
k
RS
t
RS
kt
RS
k
RSRS
t yaay
RE
t
RERS
t
RS
t
RSRS
t
RS
t
RSRS
t
RS
t hdqhqqh 1111
2
1
RS
t
RS
t
RSRS
t
RSRSRS
t hqq 1
2
11
2
1
0
k
JS
t
JS
kt
JS
k
JSJS
t yaay
JE
t
JEJS
t
JS
t
JSJS
t
JS
t
JSJS
t
JS
t hdqhqqh 1111
2
1
JS
t
JS
t
JSJS
t
JSJSJS
t hqq 1
2
11 (32)
Case 6: Impact of the short- and long-run volatility components of exogenous foreign exchange volatility in the
previous period
4
1
0
k
RS
t
RS
kt
RS
k
RSRS
t yaay
RE
t
RERE
t
RERS
t
RS
t
RSRS
t
RS
t
RSRS
t
RS
t sdqdqhqqh 1211111
2
1
RS
t
RS
t
RSRS
t
RSRSRS
t hqq 1
2
11
Asian Economic and Financial Review, 2016, 6(5): 277-297
290
© 2016 AESS Publications. All Rights Reserved.
2
1
0
k
JS
t
JS
kt
JS
k
JSJS
t yaay
JE
t
JEJE
t
JEJS
t
JS
t
JSJS
t
JS
t
JSJS
t
JS
t sdqdqhqqh 1211111
2
1
JS
t
JS
t
JSJS
t
JSJSJS
t hqq 1
2
11 (33)
4.4. Empirical Results
This article employs the BHHH algorithm to obtain the estimates of consistent estimators. Tables 10 to 15
present the estimated coefficients of conditional mean and variance equations from six cases of the three Component
GARCH-M models. It can be seen from Tables 10 to 15 that the restrictions of the coefficients (α>0, β> >0, >0
and α+β<ρ<1), are fully satisfied for all cases of the three Component GARCH-M models, indicating that all
Component GARCH-M models constructed in this study are economically meaningful.
The Ljung-Box test statistics in Tables 10 to 15 reveal that for the first and second order moments of the residual
series on Taiwanese and Japanese equity and foreign exchange markets, the distributions of these residual series are
neither auto-correlated nor hetroskedastic. These verify that all Component GARCH-M models constructed are
appropriate for our empirical analyses.
Tables 10 and 11 summarize the impacts of equity and foreign exchange volatility on returns of equity markets
and the effects of the short and long-run volatility components on both the Taiwanese and Japanese equity markets.
As indicated in Table 10, the Taiwanese equity market returns are significantly affected by lagged second and fourth
periods, implying that investors exhibit bias expectations or the information is not fully reflected. The Taiwanese
equity market may not be efficient in the sense that the investors cannot have more time to be fully adjusted when
some shocks or events occur, however current returns may be predicted by using past information. The significantly
positive coefficient RSc reveals that the Taiwanese equity market returns are also affected by the equity volatility.
The positive risk-return relation indicates a positive risk premium in the market.
The significantly positive coefficient RS of the conditional variance equation shows that the unexpected shock
of volatility will impact the fluctuations in the Taiwanese equity market. For the long-run volatility components, the
estimated coefficient RS is .957, which is significant and close to one, indicating persistency in the long-run
volatility components of the Taiwanese equity market. Finally, the significantly positive coefficient RS reveals that
the long-run volatility components of the Taiwanese equity market will fluctuate when unexpected shocks of
volatility arrive on the market.
Similar results are found in Table 10 for the Japanese equity market except for the insignificant coefficient JSc ,
which shows that there is no significant positive risk-return relationship in the Japanese equity market due to possible
biases in investor’s expectations (Fu et al., 2011).
Asian Economic and Financial Review, 2016, 6(5): 277-297
291
© 2016 AESS Publications. All Rights Reserved.
Table-10. Impact of endogenous equity volatility: Case 1
Conditional Mean Equation Conditional Mean Equation
Taiwan Japan coefficient t-value coefficient t-value
RS
0a -0.001* -1.870 JS
0a
-0.012 -1.612 RS
1a 0.018 1.032 JS
1a -0.043* -1.835
RS
2a 0.062** 2.615 JS
2a -0.002 -0.114
RS
3a 0.007 0.483 JSc 5.143 1.773 RS
4a -0.053** -2.342 RSc 6.585** 2.855
Conditional Variance Equation Conditional Variance Equation
Taiwan Japan coefficient t-value coefficient t-value
RS 0.052*** 3.502 JS 0.059*** 3.667 RS 0.284 1.214
JS 0.212 0.607 RS 0.001*** 7.084 JS 0.004*** 7.554 RS 0.961*** 140.003 JS 0.980*** 184.115
RS 0.083*** 12.115 JS 0.081*** 10.215
)24(Q 14.339 )24(Q 17.524 )24(2Q 21.492 )24(2Q 24.152
Note: *, ** and *** represents significant at 10%, 5% and 1% level, respectively.
Table-11. Impact of exogenous foreign exchange volatility: Case 2
Conditional Mean Equation Conditional Mean Equation
Taiwan Japan
coefficient t-value coefficient t-value RS
0a 0.001 1.154 JS
0a
-0.001 -0.203
RS
1a 0.031 1.265 JS
1a -0.042 * -1.817
RS
2a 0.051 ** 2.634 JS
2a -0.002 -0.071
RS
3a 0.021 0.571 JEc 0.901 0.181
RS
4a -0.062 ** -2.241
REc 0.181 0.046
Conditional Variance Equation Conditional Variance Equation
Taiwan Japan
coefficient t-value coefficient t-value RS 0.0113 0.824 JS 0.0512 *** 3.477
RS 0.831 *** 2.820 JS 0.285 0.867
RS 0.001 *** 3.815 JS 0.002 *** 7.555
RS 0.982*** 121.004 JS 0.981*** 200.116 RS 0.067 *** 4.262
JS 0.081 *** 9.943
)24(Q 17.446 )24(Q 18.343
)24(2Q 22.110 )24(2Q 21.470
Note: *, ** and *** represents significant at 10%, 5% and 1% level, respectively.
We find very similar results in Table 11 except for the foreign exchange exposure. The insignificant REc and
JEc in Table 11 indicate that both the Taiwanese and Japanese equity market returns are not affected by their foreign
exchange exposure. This is somewhat surprising. One of the possible reasons is that the risk of foreign exchange rate
consists of short-term risk (accounting risk) and long-term risk (economic risk) (Chiang et al., 2000). Thus, the short-
Asian Economic and Financial Review, 2016, 6(5): 277-297
292
© 2016 AESS Publications. All Rights Reserved.
and long-term effects will not be fully reflected if short- and long-term risks are not decomposed appropriately in the
models.
Table-12. Impact of the short- and long-run volatility components of endogenous equity volatility: Case 3
Conditional Mean Equation Conditional Mean Equation
Taiwan Japan
coefficient t-value coefficient t-value
RS
0a -0.009* -1.982 JS
0a
-0.001* -1.846
RS
1a 0.021 0.993 JS
1a
-0.042** -2.016
RS
2a 0.043** 2.235 JS
2a
-0.021 -0.690
RS
3a 0.046* 1.928 JSc1
9.142*** 5.012
RS
4a -0.068*** -3.412 JSc2 2.415 1.542
RSc1 1.701 0.281
RSc2 29.112*** 2.995
Conditional Variance Equation Conditional Variance Equation
Taiwan Japan
coefficient t-value coefficient t-value
RS 0.0612 *** 3.401 JS 0.053*** 3.411
RS 0.547 1.412 JS 0.681 1.281
RS 0.0122 1.063 JS 0.001*** 7.532 RS 0.912 *** 101.703 JS 0.982 *** 199.336 RS 0.071 *** 11.813 JS 0.092 *** 11.333
)24(Q 18.336 )24(Q 19.221
)24(2Q 22.113 )24(2Q 20.778
Note: *, **and *** represents significant at 10%, 5% and 1% level, respectively.
Tables 12 and 13 summarize the impact of the short and long-run components of equity and foreign exchange
volatility on the rate of returns and the effects of the short and long-run volatility component of both the Taiwanese
and Japanese equity markets. We focus on exploring the impacts of the short- and long-run components of equity and
foreign exchange volatility on returns of both markets. These results are very similar to those shown in Tables 10 and
11. The insignificant RSc1 and significant
RSc2 in Table 12 indicate that there is significant positive risk-return
relation and risk premium in the short-term on the Taiwanese equity market; in the long-term, the equity risk is not
fully reflected, possibly because there is a trading price limit and investors and enterprises take advantage of financial
instruments to hedge their risks or there are more individual investors in Taiwanese equity market who are irrational,
or institutional investors are less risk adverse (Barber et al., 2009; Chuang and Susmel, 2011).
On the other hand, the significant JSc1 and insignificant
JSc2 indicate that there is significant positive risk-return
relation and risk premium only in the long-term on the Japanese equity market. The possible reason is that the
proportion of financial institutional investors is large enough that only in the long-term will the equity market
generate structural change and greater fluctuations, and hence require more risk premium.
Asian Economic and Financial Review, 2016, 6(5): 277-297
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© 2016 AESS Publications. All Rights Reserved.
Table-13. Impact of the short- and long-run volatility components of exogenous foreign exchange volatility: Case 4
Conditional Mean Equation Conditional Mean Equation
Taiwan Japan coefficient t-value coefficient t-value
RS
0a -0.003 -1.032 JS
0a
-0.001 -1.124
RS
1a 0.032 1.314 JS
1a -0.031 -1.613
RS
2a 0.058** 2.512 JS
2a -0.004 -0.171
RS
3a 0.021 0.665 JEc1 13.117 1.463
RS
4a -0.051** -2.245 JEc2 -32.812** -2.545
REc1 18.554 1.062
REc2 -8.912 -0.915
Conditional Variance Equation Conditional Variance Equation
Taiwan Japan
coefficient t-value coefficient t-value RS 0.012 0.656 JS 0.055 *** 3.412
RS 0.837** 2.136 JS 0.241 0.889
RS 0.000156 *** 2.931 JS 0.002*** 8.247 RS 0.982 *** 96.298 JS 0.982*** 198.996
RS 0.061*** 2.723 JS 0.081 *** 9.994
)24(Q 18.493 )24(Q 18.562 )24(2Q 20.151 )24(2Q 21.403
Note: ** and *** represents significant at 5% and 1% level, respectively.
Table-14. Impact of exogenous foreign exchange volatility in previous period: Case 5
Conditional Mean Equation Conditional Mean Equation
Taiwan Japan
coefficient t-value coefficient t-value RS
0a 0.0013 1.381 JS
0a -0.001 -0.142
RS
1a 0.0211 1.422 JS
1a -0.031 -1.273
RS
2a 0.0521 *** 2.613 JS
2a -0.001 -0.083
RS
3a 0.0112 0.713
RS
4a -0.061 *** -2.282
Conditional Variance Equation Conditional Variance Equation
Taiwan Japan
coefficient t-value coefficient t-value
RS 0.004 0.898 JS 0.051 *** 3.241
RS 0.971*** 212.531 JS 0.283 0.813
RS 0.001*** 2.831 JS
0.002 *** 6.815
RS 0.991 *** 118.652 JS 0.874*** 190.335
RS 0.072 *** 8.714 JS 0.078 *** 8.815
RSd 0.020 *** 2.715 JSd -0.010 -0.734
)24(Q 17.141 )24(Q 19.114
)24(2Q 24.323 )24(2Q 21.497
Note: *** represents significant at 1% level.
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Table-15. Impact of the short- and long-run volatility components of exogenous foreign exchange volatility in previous period: Case 6
Conditional Mean Equation Conditional Mean Equation
Taiwan Japan coefficient t-value coefficient t-value
RS
0a 0.001 1.413
JS
0a
-0.001 -0.272
RS
1a 0.031 1.246 JS
1a -0.051* -1.743
RS
2a 0.051** 2.644 JS
2a -0.002 -0.073
RS
3a 0.015 0.743
RS
4a -0.046** -2.415
Conditional Variance Equation Conditional Variance Equation
Taiwan Japan coefficient t-value coefficient t-value
RS 0.051 1.514
JS 0.051*** 3.243
RS 0.921*** 58.342 JS 0.287 0.816
RS 0.001*** 3.489 JS 0.001*** 7.876
RS 0.993*** 126.314 JS 0.981*** 185.493
RS 0.018 0.725 JS 0.081*** 9.999
RSd1 -0.271*** -3.414
JSd1 -0.010 -0.813
RSd2 0.081** 2.164 JSd 2
-0.004 -0.064
)24(Q 17.334 )24(Q 19.114
)24(2Q 24.816 )24(2Q 20.412
Note: *, ** and *** represents significant at 10%, 5% and 1% level, respectively.
However, in the short run, the risk-return tradeoff in Japanese equity market may be caused by the bias
expectations of investors, or investors simply do not react to the new information quickly (Tai, 1999; Chiang and
Yang, 2003; Lin et al., 2009). The impacts of the short- and long-run components of foreign exchange volatility on
returns of both markets are reported in Table 13. As shown in Table 13, the insignificant REc1 and
REc2 indicate that
returns of the Taiwanese equity market are not affected by its foreign exchange exposure in either the short- and long-
run, possibly since investors and enterprises employ financial instruments to hedge their risks (Yau and Nieh,
2006;2009). The significantly negative JEc2 reveals that returns of the Japanese equity market decreases when foreign
exchange risks increase in the short-term. A possible reason is that a large proportion of financial institutional
investors possess a relatively high tolerance for risk taking in the short-term (Yau and Nieh, 2006;2009). Compared
with the results of case 2, the decomposition of the short and long-run components of foreign exchange volatility
facilitates reflecting the relative impact of foreign exchange exposure on returns of the Japanese equity market. This
result is supported by similar results from Pan et al. (2007) and Tai (2007).
Tables 14 and 15 report the results of the impacts of foreign exchange volatility in previous periods on
contemporaneous volatility. They also show the effects of the short and long-run volatility components on both equity
markets. We focus on investigating the impact of foreign exchange volatility in previous periods on contemporaneous
volatility. Other results are also similar to those found in Tables 10 and 11.
It can be seen from Tables 14 and 15 that the contemporaneous volatility will be significantly affected by the
short and long-run foreign exchange exposure in previous periods of the Taiwanese equity market, whereas the
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© 2016 AESS Publications. All Rights Reserved.
contemporaneous volatility will not be affected by the short and long-run foreign exchange exposure in previous
periods of the Japanese equity market. The lagged response of foreign exchange exposure in previous periods on the
volatility of the Taiwanese equity market possibly reflects relatively greater foreign exchange rate risk for the large
trading volume of imports and exports that characterizes Taiwan’s economy (Wu, 1997; Yau and Nieh, 2006;2009;
Liu and Tu, 2012). This also confirms that there exists a cross-volatility spillover effect on the Taiwanese equity
market, including the short-term accounting and the long-term economic risks. Moreover, the insignificant response
of lagged foreign exchange exposure on the Japanese equity market possibly demonstrates that the Japanese economy
relies more on domestic demand and thus has a lower degree of the risk of foreign exchange rate (Choi et al., 1998;
Homma et al., 2005; Yau and Nieh, 2006;2009).
5. CONCLUSIONS
Recognizing the rising interest of financial econometrics on the investigation of the correlation between returns
in equity markets and the foreign exchange rates on foreign currency markets, we employed the bivariate Component
GARCH-M model. The analysis was based on Taiwan and Japan weighted equity index and exchange rate to analyze
the risk premium for both Taiwanese and Japanese equity markets. We further explore the permanent and transitory
effects of the short and long-run volatility components on both equity markets, and thus verify the effectiveness of the
model.
We find that through decomposing the time series effects of volatility into short and long-run components, the
conditional covariance trend (long-run) and transitory (short-run) components can be studied separately with a clear
improvement in the model’s performance due to its richer dynamics that makes it easier to jointly obtain the long and
short-term market returns.
For the returns of both Taiwanese and Japanese equity markets, the empirical results are significantly affected by
lagged periods, implying that both markets may not be fully efficient in the sense that the investors do not react
quickly to shocks or new events. Thus, returns are likely to be predicted by using past information and the unexpected
shock of volatility might in general influence fluctuations in both equity markets. In addition, persistency on long-run
volatility components from both markets was also found.
The positive risk-return relation indicates that there is risk premium in the Taiwanese equity market. On the other
hand, there is no significant positive risk-return relation in the Japanese equity market if the impacts from the short
and long-run volatility components are not decomposed.
Taking into account the short and long-run volatility components, the Taiwanese equity market exhibits positive
risk-return relation and risk premium only in the short-run. In the long-run, the equity risk is not fully reflected
possibly because there is a trading price limit and investors and enterprises take advantage of financial instruments to
hedge their risks. Furthermore, there are more individual investors in Taiwan equity market who are irrational or
institutional investors who are less risk adverse. On the other hand, the Japanese equity market exhibits positive risk-
return relation and risk premium only in the long-run, implying that the large proportion of financial institutional
investors may generate structural change and greater fluctuations, and hence require more risk premium in the long-
run. However, in the short-run, the risk-return tradeoff in the Japanese equity market may be caused by the bias
expectations in investors, or because they simply do not react quickly to new information.
The impact from short and long-run components of the foreign exchange volatility on returns from both markets
indicate that returns of the Taiwanese equity market are not influenced by its own foreign exchange exposure in either
the short or long-run. Possibly because investors and enterprises employ financial instruments to hedge their risks or
there are more individual investors in the Taiwanese equity market who are irrational. Or in other cases, institutional
investors may be less risk adverse. However, returns in the Japanese equity market decreases when the foreign
Asian Economic and Financial Review, 2016, 6(5): 277-297
296
© 2016 AESS Publications. All Rights Reserved.
exchange risk in the short-run increases. Possibly since large proportions of financial institutional investors possess
relatively high tolerance on risk taking in the short-run or as the expected volatility risk rises. This might lead the
market uncertainty to increase, and the rational investors (speculators) might need incremental risk premium to
induce them to arbitrage away the mispricing. The decomposition of the short and long-run components from foreign
exchange volatility better detects the relative impact of foreign exchange exposure on returns of the Japanese equity
market.
Finally, the contemporaneous volatility is significantly affected by the short and long-run foreign exchange
exposure in previous periods of the Taiwanese equity market. The difference in characteristics of Taiwan and Japan’s
economic model may play a part in their respective exchange rate risk. While Taiwan’s exports account for roughly
60% of its GDP, Japan can potentially weather fluctuations in exchange rates by relying on domestic consumption
since its ratio of domestic consumption as a percentage of its GDP is above 55%.
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